4 problems

For every finite simple graph $G$, if its ordinary domination number equals its eternal domination number, then it equals its upper domination number:…

Given a finite graph $G$ with no isolated vertices and domination number $\gamma(G)=2$, determine exactly which graphs satisfy $\gamma_t(G)=2$; equivalently, characterize all finit…

For every nontrivial planar graph $G$, the bondage number satisfies $b(G)\leq \Delta(G)+1$, where $b(G)=\min\{\lvert F\rvert:F\subseteq E(G),\ \gamma(G-F)>\gamma(G)\}$ and…

Let $G$ be any graph, and let $M(G)$ denote its middle graph. Write $\gamma_p(M(G))$ for the power domination number of $M(G)$ and $\gamma^{'}(G)$ for the edge domination number of…